studies of injection into naturally fractured reservoirs.pdf

7/29/2019 studies of injection into naturally fractured reservoirs.pdf

1/6

STUDIES OF INJECTION INTO NATURALLY FRACTURED RESERVOIRSCudmundur S. Bgdvarsson and Cheng Hsien Lai

4Ear th Sciences Div is isonLawrence Berkeley LaboratoryUn iv e r si ty o f C a l i f o r n i a

Berkeley, Ca li for ni a 94720

ABSTRACT Warren and Root (196 31, where t h r e e or th og on alf r a c t u r e se t s of equal spac ing ar e used . Simi lartu re s a re report ed by Lauwerier (19551, Bodvarsson(19691, and Bodvarsson and Tsang (1982). Theobj ec t ive of the presen t work i s t o e x t en d t h e i rwork t o i n cl u de t h e e f f e c t s of v e r t i c a l f r a c t u r e sand t o develop a methodology fo r t he desig n ofin j ec t ion schemes fo r na t ura l l y f rac ture d geo-

A semi-analy t ica l model fo r s tu d ie s o f co ld work on non-isothermal f low in hor izo nta l frac -v a t e r i n j e c t i o n i n t o n a t u r a l l y f r a c t ur e d reser-voirs has been developed .t o d es ig n th e f lo w r a t e s and lo c a t io n o f i n j ec t io nwel l s in such systems. The res ul t s obtained usingth e model show th a t i n i t i a l l y t h e co ld v a t e r willmove very rap idl y through t he fr ac tu re system awayfrom the well. L a t e r o n , co n d u ct iv e h ea t t r an s f e r t h e rm a l r e se r v o i r s .from the rock matrix blocks w i l l r .etard theadvancement of th e col d wate r fr on t, and event- BASIC MODELually uniform energy sweep conditions w i l lp r e v a i l .ar e reached th e cold v at er movement avay from th ei n j e c t i o n well w i l l b e i d e n t i c a l t o t h a t i n apor ous medium; co ns eq ue nt ly maximum en er gy re-covery from the rock matr ix w i l l b e a t t a in ed .t i m e of uniform energy sveep and the r ad ia ld i s t a n ce f rom th e in j ec t io n well where i t occursa r e g r ea t l y d ep en dent upon t h e f r ac tu r e sp ac in g,bu t independent o f the f ra c t ure

The model can be used

Where uniform energy sveep condi t ions Ihe model used in t h i s s tudy is shown i nFigure 1.blocks bounded by th re e se t s of orthogonalf r ac tu r e s . S t ead y s t a t e f lu id f lo w i s assumed i nt h e f r a c t u r e s , b ut f u l l y t r a n s i e n t c o nd u ct i ve he a ttr an sf er between the impermeable rock matrix andt h e f r a c t ur e s i s considered. Thus, t he cold waterw i l l f low f rom the in jec t ion w e l l i n to th e f r ac-tu re network, and as i t moves away from t h e e l l ,i t w i l l gradually get heated up due t o t h e h ea tt r ans fe r f rom ad jacen t matr ix b locks.

It co n s i s t s of r ec t an g u la r m a t rix

The

* INTRODUCTION The equat ion for conduct ive hea t t ran sfe rVarious the ore t ic a l s t ud i es have shown tha t be tveen the matr ix b locks and the f ra c tu res isder ive d based on th e ba si c element shown in Figure2. The bas ic e lement re pre sen ts 1 /6 of a s in g lecube i s cons ider ed (1-Dimensional approach). Int h i s app ro ach we assme t h a t t h e th er ma l g r ad ien t sar e much smal ler wi th in t he f ra c tu re ne twork . thanin the rock matr ix . Thus, i f the tempera ture inthe f ra c tu res bounding a matr ix b lock i s r a t h e r

in je c t ion of water i n t o geo thermal re ser vo i rsd u r in g ex p lo i t a t i o n ' can g r ea t ly en hance th eenergy recovery from t h e resource . In je c t - matr ix b lock (cube) , i . e . only one face of th eio n vi11 h e lp m a in ta in r e se r v o i r p r e ssu r e andprovide water t h a t w i l l ex trac t energy f rom there se rv oi r rocks. However, experienc e gainedthr ou gh co m er c i a1 s ca l e r e in j ec t io n h as shownth a t t h e i n j ec t i o n o p e r a t io n mus t b e ca r e f u l lydesigned.o f p r od uced f l u id s a t sev e r a l Jap anese f i e l d sd ue t o i n j e c t i o n . T hi s i n t e r f e r e n c e i s a t t r i b u t e dt o r a pi d flow of the i nje ct ed "cold" water throughf r ac t u r e s . a r evidenced b y h ig h t r ac e r v e lo c i t i e s .On the o t her hand commercial s ca le in j ec t ion hasb een su ccess f u l a t o th e r g eo th er ma l f i e l d s su ch asthe Geysers, U.S. and Ahuachap&, E l Salvador .This ind ica tes tha t fundamenta l s tud ies of co ldwater movement in f rac tur ed geothermal res er voi rsar e needed , before conf idence in t he designof commercial re in je ct ion operat ions can bee s t ab1 shed.

Horne (1981) r epor ts en tha lpy dec l in e

In th is s tudy the prob lem of co ld wateri n j e c t i on i n t o n a t u r a l l y f r a c t u r e d r e s e r v o i r s isconsidered. The basis model used is chat of (1963).

Figure 1. Ideal ized model of n a t u r a l l y f r a c t u r e dre se rv oi rs ( a f te r Warren and Hoot


2/6

Bodvarsson and L a i22n = -D

(4c 1MotrirBlock4pfcfbe =- 'rcrH

BasicElement Th e r ea l p a r am e te r s i n eq u a t io n s 4 a r e de f i ne d i nthe Nomenclature.The sol ut io n of eq uat io ns (1-3) can be ob-

ta ine d in the La place domain as (Bodvarsson andLei, 1982):igu re 2. 1-Dimensional approximation of hea tconduction in rock matr ix b locks.

uniform, a heat conduct ion equat ion based on t hebas ic e lement (Fig . 2 ) is a reasonable approxima-t i o n . W have ve ri f i ed t h i s approach by comparingour model t o a 3 4 i m e n s i o n a l s o l u t i o n ( L a i andBodvarsson, 1982). The approach des cri bed aboveis qu i t e s im i la r t o the one employed in the"Mul tipl e In te ra ct in g Continuum" Method (HINC)for numer ica l model ing of hea t and f lu id f low infractured porous media (Pruess and Narasimhan,1982).

Equat ions ( 5 ) and ( 6 ) are inver ted using a nmer-i ca l method developed by St eh fe st (1970).

RESULTSIt is o f pr im ar y i n t e r e s t i n t h i s s tu dy t oexamine th e ra t e with which the "cold" water f ron tadvances away from t h e i n j e c t i o n w e l l dur ing

i n j e c t i o n . lh is in format ion i s u s e f u l i n t h ed es ig n of t h e sa f e l o c a t io n an d ra tes o f in j ec t io nve l l s i n r e l a t i o n t o t h e p r o d u c t io n w e l l s . Ww i l l d e f in e th e the r m a l f r o n t ( co ld wa ter f r o n t )as t h e lo cu s o f p o in t s w i th t em p er a tu r e b ein g th eav e rag e o f t h e i n i t a l t em p e ra tu r e o f t h e r e se r -voir (To) and, th e temperature of t h e i n j e c t e dwa te r T i ( Tm = l/P[To+Ti]).d i me n si o nl e ss r a d i a l d i s t a n c e 6 of the thermalf r o n t from t h e well i s plo t te d aga inst d imension-l e ss time T f o r v a r i o u s v a l u e s of 6 . The para-meter 8 r e p r e s e n t s t h e r a t i o o f t h e a v e ra g e fr a c -t u r e a p e r t u r e ( b ) t o t h e r e s e r v o i r t h i ck n es s ( H) .In most c a se s r e a l i s t i c v a lu e s of 0 l i e i n t h e

I n Figure 3 t h e

UATHEMATICAL DEVELOPMENT' In dimens ionless form the governing equa-t io n s f o r t h e t em p era tu r es i n t h e f r ac t u r e s ( T2)and the rock matr ix (TI) a re :F r a c t u r e s (n = 1):

l o * , , , , , , , , ,

(3b)

(3c)

The dimensionles s parameters in equa tions (1-3)a r e d e f in ed a s :T - T. & A t T

Figu re 3. Type curv es fo r thermal f ron t movementi n n a t u r a l l y f r a c t u r e d r e s e r v o i r s .

T =- 2D =0 (4.)Ti - To 1 " rCrD2


3/6

'range of 10-5 - 10-8.f o r a given value of t ) , t h e r a d i a l l o c a t i o n of t h et li ar lnal f ront (co l d water f ront in the f r ac tur es )away froni the inject ion w e l l , at any given time.I f on e follows the advancement of the thermalf ron t fo r one va l ue o f e , s a y e = 10-5, one canse e t h re e d i f f e r en t r a t e s o f advancement . A tea r l y t imes when conduct ion heat t r an sf er f rom therock matrix is neg l i g i b l e , t he f ron t moves asrZ/t away from the in jec t io n wel l . During th i sper iod th e thermal f ron t moves in the f ra ctu reson ly , in an analogous manner t o a s i n g l e r a d i a ls y s t e m with insulated upper (caprock) and lower(bedrock) bound aries , and a thi ck ne ss correspond-ing t o t h e average f r ac t u re ape r t u re . B odvars son(1972) has der ived an exp ress ion fo r t he movementof t he t he rma l f ron t i n t h i s case. A t i n t e r -m edi a te t i m es , t he s l op e i n F i gu re 3 d e c r e a s e sby ha lf , and conseque ntly th e advancement of th ethermal f ront is p r o p o r ti o n a l t o r 4 / t . Duringt h i s pe r i od t h e conduc ti ve hea t t r a n s f e r be tweent he rock m a t r i x and t h e f r ac t u r es domi nat es, re-s u l t i n g i n a much slower movement of t he t her malfront away f rom the in ject ion w e l l . The largeh e a t t r a n s f e r c a u s es a very slow movement of t h et herm al f ron t i n t he f r a c t u res , bu t r ap i d ex-t r a c t i on o f hea t f rom t he rock m a t r i x

The fi gu re ac tu al ly sho;s

F i n a l l y , a t v e ry l a t e t i m e s (I 1. 1.01, t h et he rm a l f ron t aga i n advances a t a r a t e p ro po r-t i o n a l t o d / t .hea t t r a n s f e r be tween rock m a t r i x and t h e f r ac -t u r es ha s been r eached , and cons equen t l y t het he rm a l f ron t w i l l move as i f o nl y 8 porous mediumwas pres en t i.e. i ndependen t o f t he f r ac t u r ena t u re o f t he r e s e r vo i r . Kowever, i n con t r a s t t ot h e e a r l y time behav i o r , t he t he rm a l f ro n t nowmoves at t h e same r a t e i n t h e f r a c t u re s as i n t h erock mat r ix .

A t t h i s t i m e q u a s i - s t e a d y s t a t e

I n orde r t o e x p l a i n t h i s more f u l l y F i g u r esThe figures show the t ime-5 were cons t ruc t ed .

sequence of the d imens ionless t emperature pr of i l esaway from the inject ion wells i n t h e f r a c t u r e s a ndt he rock m a t r i x fo r a g i ven va l ue o f 8 . Thedimens ionless t emperature of TD = 1.0 r e p r e s e n t st he t em pera t u re o f t h e i n j ec t e d wa te r , whereas t h edimens ionless t emperature TD = 0.0 correspondst o t h e i n i t i a l r e s e r v o i r t e mp e r at u re . T em pe ra tu rep r o f i l e s a r e g iv en f o r t h e f r a c t u r e s (n = 1.01,t h e c e n t e r o f t h e c ub es (n = 0 . 0 ) and tvo in ter-mediate values (n = 0.3, 0.7). The f igures show

cFigure 4. Tempera tu re p ro f i l e s i n f r ac t u re s and

rock mat r ix ; I = 0.1.

Figure 5 . Tempera tu re p ro f i l e s i n f r ac t u re s ' androck mat r ix ; t = 1.0.

t h a t a t e a r l y times (T = 0 . 1 ) t he re i*s a con-s i de rab l e d i f f e r ence bet ween t he t em perat u re p ro -f i l e s i n t h e f r a c t u r e s a nd t h e r o c k m a tr i x. A tl a t e r times t h e c u r v e s s t a r t t o c on ve rg e, a l th o ug ht he co l d wa t e r f ro n t is co ns ta nt ly moving awayfrom the well. A t a dimens ionless t h e ofT = 1.0 t h e t e m p e r a t u r e p r o f i l e s are p r a c t i c a l l yi den t i ca l i n t he f r ac t u re s and t he rock m a t r i x .This can be shown ana ly t ic al ly by cons i der ing ana s y mp t ot i c s o l u t i o n f o r t h e l a t e time behav i o r o fequa t i on ( 6 ) .

The reas on f or t h i s phenomenon i s t h a t a te a r l y time t he co l d wa t e r s hoo t s . r ap i d l y t h rought h e f r a c t u r e s , i n c r e a si n g th e s u r f a c e area f o rconduc ti ve hea t t r a n s f e r be tween t he f r ac t u r es andt h e rock m a t r i x . The l a rg e s u r fac e area enhancesenergy t r a ns f e r f rom t h e rock m a tr i x t o t h e f r a c -t u re f l u i d s , t hus r e t a rd i ng t h e advancem en t o f t hec o l d w a te r f r o n t a lo n g t h e f r a c t u r e s . T h i s i nt u rn , t ends t o e q u i l i b r a t e t h e t e m p er a t ur e s i n t h ef r a c t u r e s a nd t h e r o c k m a t r i x . s o t h a t e v e n t u a l l ythe temperature prof i l es away f rom the w e l l arei d e n t i c a l f o r t h e f r a c t u r e s and t h e r o c k m a t r ix .

PRACTICAL CONSIDERATIONSThe most i n t e r e s t i n g a s pe c t o f t h e r e s u l t s

obta ined is t h a t e v en f o r f r a c t u r e d r e s e r v o i r s ,uniform energy sweep of the r es er voi r can beobtained . A uniform energy sweep w i l l maximizet h e amount of recoverable energy from the re-source . A necessa ry requirement for such condi-t i o n s is t h a t t h e i n j e c t i o n w e l l s b e a p p r o p r i a t e l yl oca t ed wi t h r e s pec t t o the product ion wells.S i m i l a r conc l u s i ons were obtained by Bodvarssonand Tsang (1982) for th e case o f h o r i z o n t a lf r a c t u r e s o nl y ; howe ve r i n t h a t c a s e t h e c r i t e r i af o r p ro pe r s i t i n g o f t h e i n j e c t i o n vel ls aredi f f er en t f rom what w e propos e he re fo r na t u ra l l yf r a c t u r e d r e s e r v o i r s .

For t he des i gn o f an i n j ec t i o n sy st em fo rna t u ra l l y f r a c t u red r e s e r vo i r s m a themat ica lexp res s i ons t ha t can be us ed t o ca l c u l a t e t he t i m eand r ad i a l d i s t a nce f rom t he i n j e c t i on wells whereuni fo rm s weep cond i t i ons p re va i l a r e qu i t e u s e fu l .F igure 4 shows that uniform energy sweep condit ionw i l l prevai l when:c (& + e ) = 1 = 1.0 ( 7 )In genera l 4 >> 0 so t ha t equa t i on ( 7 ) can bewr i t t en in t erms of rea l parameters as :

3


4/6

Bodvarsson and Lai

Inspec tion of equati ons (8s) and (8b) shows th atbo th the t ime and ra d ia l d is tan ce of t he un iformenergy sweep condition depend g re a tl y on th ef r ac tu r e sp ac in g D .independent o f the f ra c tu re aper tu re b . Frac ture sp o sse ss ing sm a l l ap e r tu r e s w i l l co n ta in v e r y sm a l lamounts of f luids, so tha t even though f lu idv e l o c i t i e s a r e hi g h ve r y l i t t l e energy i s neededt o increase the tempera ture .

However, both qua nt it ie s are

Now l e t u s con s id e r an in j ec t i o n wel l i n an a t u r a l ly f r ac tu r ed r e se r v o i r u s in g t h e p ar am e te rsshown i n Tabl e 1. I f t h e av e r ag e f r ac tu r e sp acin gis not known the following exp ressi ons can beca lcu la ted .r = 2.6 x D (meters) (9s)

(9b)2t c 0.01 x D ( y ea r s )Thus, fo r an average frac tu re spacing of 50meters, uniform energy sweep conditions w i l lp r ev a i l 1 3 0 m away from t h e w e l l a f t e r 2 5 y e a r sof i n j e c t i o n .Table 1: Par ame ter s Used i n ExampleI n j e c t i o n ra te , q : 20 k g l sFlu id d en s i ty , pw : 1000 kg/m3Flu id h ea t c ap ac i ty , cy ' : 4200 J/kg.*CThermal conductivity, X : 2. 0 J/m.s.'CReservoir th ickness , H : 500 mRock density, pr : 2700 kg/m3Rock heat capacity, cr : 1000 J/kg.'C

I n summary, we have developed a model th at can beused t o determine proper loc ati ons and f low ra teso f i n j e c t i o n wells i n n a t u r a l l y f r a c t u r e d reser-v o i r s .j e c t i o n o p e ra t i o n so that maximum energy recoveryfrom the resource is obtained.

The model can hel p i n opti mizin g an in-

NOMENCLATUREb:D :H:I112,P :9 :q:r:rc :

t C:t :T:

Fr ac tu r e ap e r tu r e (m )Fracture spac ing (m)Reservoir thickness (m)1312: modified sp he ri ca l Bessel func-t i o n s of t h e f i r s t ki ndLaplace parameterPo r o s i ty (-1I n j ec t io n r a t e ( m 3 / s )Radial coordinate (m)Radia l d is t ance from in jec t io n well t o t heloc ati on of uniform energy sweepTime (seconds)Time of uniform energy sweep (s eco nds )Temperature ('C)

Ti : In je c t io n tempera ture ('C)To: I n i t i a l res erv o ir tempera ture ('C)TTF: The isotherm define d as th e " thermal front" ,u: - Temperature in fr ac tu re in Laplace domainv: Temperature in rock mat rix in Laplace domaint : Rock matrix coordinate (m)A:oc: Vol unet ric he at ca pa ci ty (J/m3.'C)Subscr ip ts :

TT F (Ti + To)/2

Thermal cond uct i v i ty o f rock matr ix(J/m. s 'C)

f : F r a c t u r er: Rock matrixw: Liquid water

ACKNOWLEDGEMENT

C r i t i c a l r evi ew o f t h i s m an u scr ip t byS. Benson, C. Doughty, H. Lippmann and K. Pruessis most apprec ia ted . This work was supported byth e Ass i s t an t Sec r e t a r y f o r C o n se r v a t io n an dRenewable Energy, Of fi ce of Renewable Technology,Division of Geothermal and Hydropower Technolo-g i e s of t h e U.S. Department of Energy underContract No. DE-AC03-76SF00098.

REFERENCESBodvarsson, C., On the temperature of water f low-

ing th rough f rac tures: J . Geophys. Res.,Vol. 74 , No. 8 , 1969.Bodvarsson, C. , Thermal p rob lems i n the s i t i ng ofr e i n j e c t i o n wells: Ceothermics, Vol. 1,NO. 2, 1972.Bodvarsson, C.S., and Tsang, C.F., In je ct io n andthermal b reak through in f ra c tu red geo thermalr e se r v o i r s : J . Ceophys. Res., Vol. 87, No.82, pages 1031-1048, 1982.n a t u r a l l y f r a c t u r e d r e s e r v o i r s ; p ap er i nprepera t ion , 1982 .Japan: paper p resen ted a t the Cal i fo rn i aRegional SPE meet ing, Bak ers fie ld, Cali -f o r n i a , SPE-9925, (March 25-26) 1981.L e i , C.H., and Bodvarsson, C.S., : Ve ri f i ca ti onof t h e HINC method, in pre par ati on, 1982.

Lauwerier, H.A . , The t ran spo r t o f hea t in an o i llayer caused by the i n je c t i on of ho t f l u id :Appl. Sci. Res. S c a t . A . , 5 , 145, 1955.method f or modeling fl ui d and heat flow infra ctu red porous media: paper presented a tthe 6th annual meeting of SPE in N e w Orleans,SPE-20509, (J an . 31-Feb. 21, 1982.tra nsf orm s: Communications of ACM, Vol.13, p. 44-49, 1970.

Warren, J . E . , and Root, P. J., The behav ior of nat-u r a l l y f ra c t u r e d r e s e r v o i r s : S o c ie t y ofPetroleum Engineers Journal, September 1963,pp. 245-255.

Bodvarsson, C.S., and La i, C.H., I n j e c t i o n i n t o

Horne, R. , Geothermal re i n je c t ion exper ience i n

Pruess , K., and Narasimhan, T.N., A p r a c t i c a l

S t e h f e s t , H., Numerical inversion of Laplace

4


5/6

This report was done with support from theDepartment of Energy. Any conclusions or opinionsexpressed in this report represent solely those of theauthor@)and no t necessarily those ofThe R egents ofthe University of California, the Lawrence BerkeleyLaboratory or the Department of Energy.Reference to a company or product name doesnot imply approval or recommendation of theproduct by the University of California or the U.S.Department of Energy to the exclusion of others thatmay be suitable.


6/6

studies of injection into naturally fractured reservoirs.pdf

Documents

Transcript of studies of injection into naturally fractured reservoirs.pdf